Honestly, I think that only close/immediate kin and very close family friends (ie, those friends to whom one is so close they are just like part of the family) should actually outright ask for belongings. If someone that I only knew in passing or didn't know very well at all would have come asking for things after either my Grandma or Pappaw died, I would have been taken aback to say the least.
We had a pretty good system for dividing things out when my Pappaw died. If you gave it to him or Grandma, you got it back (if you wanted it back). Anything else, we just kind of auctioned off among the family. (For example, "Oh, look at these bowls - does anyone want them?" "Bottlecaps, you gave this to Grandma & Pappaw didn't you? Do you want it back?") Grandma and Pappaw weren't exactly well-off, so they didn't have anything of great monetary value, but they had plenty of things that held sentimental value to their kids and grandkids.
The only issue I had was when my cousin professed that Grandma's commemorative plates, which had belonged to our great-grandmother, had been promised to her. Meanwhile, I knew that Grandma had promised one of them to me. (It had a Roy Rogers quote on it - "A man who doesn't make a mistake is a man who doesn't do anything." I'd always loved that quote.) However, I figured that I wouldn't make a fuss, she could take all the darn plates, because I knew what I wanted, and I knew no one else would really want what I wanted.
I took a cue from Mr. Bottlecaps, who only asked for one of his Dad's work shirts after his Dad died, and asked only for one of Pappaw's outfits, one of Grandma's outfits (Pappaw kept most of Grandma's things after she died), one of my Pappaw's winter coats, and my Grandma's pickling canister. I took the outfits because they're easy to store, as I could just hang them up in the closet, and they were actually used by my grandparents, unlike knick-knacks that just sat on the shelves. I could actually remember my Pappaw and Grandma wearing those clothes. I took Grandma's pickling canister because she made the most awesome red beet pickled eggs, and one of my favorite memories as a kid was sneaking into said canister before they were thoroughly pickled because I just couldn't resist. :-P That was actually a running joke in the family for years - my tendency to sneak into the pickled eggs before they were completely pickled, hehe. No one minded that I was taking the canister because of that. I took Pappaw's coat because when I'd visit him, I'd go outside to smoke. A lot of times, I wouldn't be wearing a good heavy coat, and he'd tell me to put one of his on. He hated the fact that I smoke cigarettes, but regardless, he didn't want me getting cold out there while I did it.
I think I have something in my eye now....